A way to prove, by experiment, that objects are attracted to one another proportional to their mass and the square of the distance between them multiplied by a universal constant. Did you know that? 220 years ago, it was hard-core science, maybe equal to CERN today (but not as expensive), but today, it's the sort of thing a high school student can do.
The general idea is that you have one mass on a rod hanging from a string (actually two), put it near another mass (actually two), and measure how much the string twists. It's a lot more complicated than that - you need to make sure there are no air currents or electrical charges, take into account the twisting that would happen anyway because it's too hard to totally stop the thing from swinging a little, make sure everything is really level, etc, and then the amount you're measuring is REALLY small for masses small enough for you to move around.
You could say "Cavendish was part of the conspiracy, he made it up," but then why can high school students - or anyone - reproduce the experiment and get the same result? I suppose then you could claim that all high school teachers are part of the conspiracy and they are nudging the experiment to fake the data... what college class tells physics teachers to do it, and how does every physics teacher in the world keep their mouth shut about it? "density" is irrelevant, because you don't measure things going up and down, gravity attracts them sideways.
learn more about the Cavendish Experiment
Search YouTube for "cavendish experiment" or even "cavendish experiment high school" and you'll find many cases of people recreating the results. The experiment has been reproduced countless times in the last 220 years - when properly done, it produces a predictable result. You could, if you wanted, build an apparatus and observe the effect of gravity in your basement. But it would be a major pain in the ass, which is why I just trust other people to do it, just like how I believe China exists even though I've never been there, and I assume you exist although you seem extraordinarily improbable.
The Cavendish experiment measures gravity as a horizontal force in a vacuum, not a vertical force in air. There is no bouyancy involved.
There are lots of other experiments to measure gravity, but that was the first one. In fact, your cell phone has a device called an accelerometer that measures any force on it, including gravity. Programmers on the iPhone can tell which way the phone is facing by looking for the 10m/s^2 force of gravity. That's how your phone knows which way is up. These accelerometers showed up on Macs first - when a Mac falls, it notices that there is no longer a gravitational force on the accelerometer, and it parks the hard drive head to keep it from damaging the platter. Of course, this would work just as well if the earth was a flat plane accelerating at 10m/s^2 "up", so that alone doesn't prove gravity exists.
But if you have a sensitive enough accelerometer, you CAN measure less gravity at the top of a mountain or in an airplane than on the ground, which would make no sense if the earth were an accelerating plane.
More recently, the LIGO (Laser Interferometer Gravitational-Wave Observatory) detector has been built, which has just three months ago detected "gravity waves" from far away in the universe. But that's a big expensive experiment (~$1.1 billion), too hard to replicate in your home right now, so from your point of view it would be part of the conspiracy, and we'll just ignore it.
Gravity is real. It is describable, measurable, and can be used to make useful predictions.
This in itself does not prove that the earth is round, except inasmuch as the fact that every bit of mass in the universe attracts every other bit of mass precludes the practicality of having a flat plane that large. But perhaps it is made of an unimaginably strong material, or perhaps it is actually a cube, cylinder, or cone and we're just on one flat face - even in the flat earth model, it has some thickness, right? So even you don't think it's LITERALLY a plane any more than a sheet of paper is. The gravity of a disk, of course, would pull us towards the center as many have argued - but if it's thicker at the edges you can cancel that out.
tl;dr Gravity is real, that doesn't prove that the earth is round. But, it's real.
The simplest explanation of the answer for this is that - at the "local" level (atomic/subatomic) gravity is NOT the STRONGEST force. The electromagnetic force is MUCH stronger. That's why a magnet can hold things off the ground, and why if you rub a balloon on your sweater it will stick to the wall instead of falling. (gravity actually is the stronger force in another sense according to some internet sources, but bear with me)
The electromagnetic force is (generally) the force that holds things together, like your feet, like a birds wings, and it's the force that keeps things apart, like between your feet and the floor, or a wing and the air. So why doesn't it always overpower gravity? Well, unlike gravity, the electromagnetic force comes in two flavors, which we call "positive" and "negative". The one attracts the other and repels its own kind, and the two cancel one another out.
if you're close to one end of a magnet, you can measure the force to that end, but if you're far away from the magnet, the two cancel each other out. Atoms have a positive part (the nucleus, with positive protons) and a negative part (the electrons) which are attracted to one another, which is why electrons orbit around the nuclear of an atom - sort of similar (ignoring quantum mechanics) to how the earth orbits around the sun.
(this begs the question of why electrons don't go crashing into the center of atoms - the answer is complicated and would distract from my point, but there is an answer. also, why don't protons repel each other and fly apart? an even stronger force, the "strong nuclear force" - again, complicated, let's not get distracted. How can molecules form at all? because they share electrons, past that, again, complicated.)
Why don't you sink into the earth? Because the electrons in your feet repel the electrons in the ground with a stronger force than gravity. How can birds fly? Because their wings push against the air, and the electrons in the air molecules push back with a stronger force than gravity. If there was less air, or their wings were smaller, or gravity was stronger, or the earth were more massive, they couldn't.
But at a distance, the positive and negative electrical charges cancel out and don't have any effect. So the sun doesn't have a significant positive or negative net electrical charge, nor does the earth, nor does the moon. The all pull at each other - in fact, to be more precise, the moon doesn't exactly orbit the earth, the two orbit around a common center ("barycenter"), but because the earth is much more massive, that common center is close to the center of the earth. Similarly, the earth pulls the sun just a little teensy bit, so small you can generally ignore it. YOU attract the earth to yourself!
In fact, the earth doesn't technically orbit the center of the sun - the barycenter for the whole solar system is a bit outside the surface of the sun, and the sun orbits around that along with the other planets - this is mostly due to Jupiter being so massive. It's close enough that we can generally ignore the difference, but it's there.
Unlike electromagnetism, gravity doesn't come in positive and negative flavors - it just attracts. Or, rather, there might be a repulsive kind, but all the "negative mass" matter repelled our matter and vice versa, so there isn't any around us. That's the sort of thing that expensive experiments like CERN might someday give us - a repulsive form of gravity, and I'll leave it to you to imagine what sorts of sci-fi tech that would provide.
If gravity pulls everything, why doesn't the earth crash into the sun? In short, because it got lucky. An "orbit" happens when something has just the right momentum and direction to almost fall into the bigger thing and miss. Imagine that we're far away from the earth and fire a bullet just to the right of earth. Ignore air, or use a bullet the size of a Cadillac. Fire too far to the left and/or too slow, and it crashes. Fire too far to the right and/or too fast, and it misses and keeps going off into space. Fire at just the right angle and just the right amount of force, and it misses, but it gets pulled left... just enough to keep missing. That is an "orbit".
Long ago, imagine that the whole solar system was a bunch of rocks of different sizes, all attracting one another with gravity. Some of them crashed into one another and stuck together... so their mass combined, and pulled in more stuff, and more and more. Maybe imagine survivors from the titanic floating in the water and linking up together to stay afloat.
tl;dr Gravity sucks on everything, everything pulls at everything else, but things push harder with electromagnetism in practical usage.
Well, you're right. Bouyancy is a real thing, and in practice on earth, it's what keeps the air above the water, the water above the earth, and makes a balloon float.
But bouyancy is an expression of gravity. You're talking about density, or mass divided by volume. This happens because the atoms (or molecules) of something "heavy" (denser) is more attracted by the earth than something "lighter". So the molecules of (liquid) water get pulled harder than molecules of air or even steam, and they go closer to the earth. Ice is less dense than water - one of only a few molecules that get let dense when they freeze, and I think the only one that can do it close to room temperature.
This only happens, though, because there's lots of those molecules, they're in constant motion ("brownian motion"), and they're "slippery" - repelling each other with the electrons at their surface.
Let's imagine, instead, that we make thousands of plastic balls, fill equal numbers with helium, some with plain air, and some with gravel. (the helium balls are still heavier than air, but lighter than the ones filled with air). Now, let's put the helium ones in a swimming pool, so they sit on the bottom. Then let's pile the ones filled with air on top of them. Finally, let's pile on the ones with gravel.
Well, they're just going to sit there. There's too much friction between the balls for the ones filled with gravel to sink. (I picked gravel because lead probably would be heavy enough to overcome the friction). The heavy ones stay on the top, the light ones stay on the bottom.
Now, let's shake the pool. Earthquake!!!! The balls shake around, the heavy ones (mostly, slowly) sink to the bottom, the light ones (slowly, mostly) rise to the top. The ones filled with air mostly stay in the middle.
This is how bouyancy works. Gravity pulls harder on things that have greater mass relative to their volume, and those things crowd out lighter things.
Remember, though, this isn't infinite. A hot air balloon will only rise until the point where air pressure is equal to the density of the hot air in the balloon. In fact, this is why there is such a thing as a "air pressure" that reduces as you go higher or "water pressure" that increases as you - gravity pulls as much air or water as it can down to the surface or ocean floor. The higher you go, the less air or water there is being pulled down on you by gravity.
...and it's a little more complex than that, which is why when you float on water you might sink halfway down, but that's the general idea.
another example you might think about is imagine filling the swimming pool with vertical cylinders of plastic/aluminum/lead, so they are tightly packed and can slide relative to one another but not get over or under one another. Shake all you want, they'll still all touch the bottom.
If you have a sensitive enough accelerometer, you can measure less gravity in places where the earth is (for whatever reasons) less dense, or more gravity in places where the earth is more dense. That would not be possible if the earth were a plane accelerating at a constant 10m/s^2 "up".
tl;dr Bouyancy IS gravity.
gravity is real. you can measure it yourself. electromagnetism is more powerful on small scales. bouyancy is gravity.
The essence of science is that you can be wrong, you can disprove things. Scientists do NOT (contrary to what you may think) come up with an idea and then think of experiments to prove that they are right. (Except in the pharmaceutical industry, they do this, but they're assholes.) Instead, they come up with an idea and they try to prove that they are WRONG, or that someone else is wrong.
I can think of a thousand experiments that COULD convince me that the earth is flat, or that there is no such thing as gravity. A failure in the Cavendish experiment, for example, or a clear explanation why it is flawed - and failures in every other related experiment, and explanations of why those are flawed.
There are no experiments that can convince you that the earth is round. Your mind is made up, and you reject any experiment that demonstrates the contrary.
I leave you with a short excerpt of a talk by the witty, brilliant and engaging
Richard Feynman, wherein he explains in brief how we first measured the speed of light. He briefly touches on some of the matters here. I encourage you, if you truly want to understand the universe, to seek out his lectures and absorb them.
I know you will attack me and deride me; I do applaud your instinct to ask questions, however misguided it might be - it's the right idea, but you're going about it the wrong way.
I hope only that this message reaches at least one uninformed person who might have been swayed to believe that the earth is flat and that gravity is a lie, to show them to think for themselves, understand the evidence, and not follow blindly something they read on the internet.
लॉका: समस्ता: सुखिनो भवंतु !